Single-particle electron cryomicroscopy (cryo-EM) is a method for observing the three- dimensional structures of large macromolecules. In cryo-EM the experimental data consist of noisy, random projection images of macromolecular particles, and the problem is finding the 3D structure which is consistent with these images. The proposal aims to develop and apply to experimental data novel algorithms for solving two difficult mathematical problems posed by this technique of structural biology. First, classical cryo-EM reconstruction techniques assume that the particles are identical. However, in many datasets this assumption does not hold. Some molecules of interest have more than one conformational state. These structural variations are of great interest to biologists, as they provide insight int the functioning of the molecule. The first area of investigation in this project is the development of algorithmic and mathematical framework for determining structures associated with heterogeneous particle populations. The proposed algorithm is not only faster than existing techniques but is also mathematically provable to reveal the different conformations if the number of images is sufficiently large. Second, a major limiting factor for present cryo-EM studies is the particle size. Images of small particles are often too noisy for existing methods to provide valid three-dimensional reconstructions; although the images contain structural information, the assignment of orientations to the individual particles is unreliable. The second area of investigation focuses on developing a radical new approach for reconstruction of small particles without the need for determining particle orientations.